8th Grade - Gateway 2

The instructional materials reviewed for Grade 8 partially meet the expectations for giving attention to conceptual understanding. Overall, the materials miss opportunities for students to make connections among standards and develop conceptual understanding of key mathematical concepts, especially where specific content standards or cluster headings call for it.

• Attention is given to conceptual understanding by developing concepts using prior knowledge of previous grades and combining that knowledge to understand functions by graphs, rate tables, equations and proportions found in chapter 3.
• Cluster heading 8.EE.B requires students to make connections between proportional relationships, lines, and linear equations. The text does not present content in a way that develops conceptual understanding because students learn about these in isolation with the exception of Activity 3-3b, questions 2 – 4. The two standards 8.EE.B.5 and 8.EE.B.6 are covered in this text four chapters apart.
• F.A.1, which states students should understand that a function is a rule that assigns to each input exactly one output and that these are ordered pairs, is taught in two different chapters (in lesson 3-2 on tables, and lesson 4-2 on graphs). Separating these lessons weakens the potential for conceptual understanding.
• For 8.G.A, the activity lab in chapter 6 on angle sums is minimal. There are attempts at conceptual understanding, but they should be revised with the MPs at the core of how students are learning mathematics.

Chapter 4 gives attention to conceptual understanding by building on prior knowledge of unit rate connecting it to slope and the use of slope to graph a proportional relationship, but drawings that show similar triangles formed by the slope are not defined or discussed.

The instructional materials reviewed for Grade 8 partially meet the expectations for giving attention to procedural skill and fluency. Overall, the materials give opportunities to practice with procedural skills when those skills are first introduced, but the materials do not give opportunities with different procedural skills throughout the year so that fluency is completely addressed and developed.

• In chapter 2, algebra tiles are introduced to solve multi-step equations. The tiles support the algorithm to solve multi-step algebra problems. Although there are examples using algebra tiles, the lessons are mostly procedural based with problems to develop the skills and fluency.
• Each lesson has a test prep and mixed review section, but many of these items (between 40 to 50 percent) are multiple choice, which doesn’t ensure fluency or knowledge of procedural skills.
• In Lesson 5-4, students are required to solve systems of equations and explain why they chose a specific method, which would enhance fluency since students are selecting the most efficient strategy and explaining.
• Solving systems of equations (8.EE.C.8.B) has practice in chapter 5, but the author does not seize the opportunity to emphasize it again in chapter 7.

For 8.EE.C.7, chapter 2 has practice for procedural skill and fluency, but the author does not seize the opportunity to emphasize it again in chapter 7.

The instructional materials reviewed for Grade 8 partially meet the expectations for giving attention to applications. Overall, the materials are designed so that teachers and students spend limited time working with engaging applications of the mathematics, without losing focus on the major work of each grade, and there are very few opportunities for students to engage in non-routine problems.

• The materials provide single and multi-step contextual problems, but there is a lack of non-routine problems.
• Each lesson has a challenge and real-world problems in the homework. The application problems do stress the major work of the grade.
• A large portion of this text is devoted to solving problems procedurally or making calculations to find a correct value with no real connection to any contextual situation. Most of the lessons have either four or five problems set in a real world context, and most of these are low level requiring nothing more than the standard algorithm.
• Only five problems out of 43 in lesson 2-2 ask students to define variables and simplify expressions for given contextual situations.
• In the beginning of the book there is a problem solving handbook, which has students explore different methods of solving problems. Many of these involve applications to the real world; however, they do not require students to apply their own Mathematical reasoning to each situation.
• For 8.F.B.4, only two out of 14 exercises in 4-1, six out of 26 in 4-2, nine out of 13 in 4-3, and three out of 22 on the chapter 4 assessment are application problems.

For 8.EE.C.8.C, only three out of 21 in 5-1, six out of 17 in 5-2, two out of 13 in 5-3, four out of 18 in 5-4, and five out of 20 on the chapter 5 assessment are application problems.

The instructional materials reviewed for Grade 8 partially meet the expectations for balance. Overall, the three aspects of rigor are not always treated together nor are they always treated separately within the materials, but there is not a balance of the three aspects of rigor within the grade.

• The materials have an emphasis on procedural skill and fluency providing numerous skill-driven problems.
• Examples of models to develop conceptual understanding are provided, and there are real-world problems provided in the homework sections. However, the materials do not create a balance between the three aspects of rigor by neglecting the development of conceptual understanding and by not providing enough non-routine problems.
• While the three aspects of rigor are not always treated together, the text lacks enough attention to conceptual understanding which creates an imbalance between all three aspects.

Examples and problems set in real-world contexts are too scaffolded to require students to apply the mathematics learned in the lesson.

The instructional materials reviewed for Grade 8 partially meet the expectations for identifying and using the MPs to enrich the mathematics content throughout the grade. Overall, the MPs are identified in different places in the materials, but the places where they are identified do not provide enough explanation as to how the practices enrich the mathematics content.

• The MP are identified on pages T26-T31.
• Each practice is defined and explained how it is used throughout the materials, such as: MP1 found in the Guided Problem Solving exercise within the homework problems; MP3 and Error Analysis; and MP5 found in the Activity Labs and Choose a Method exercises.
• The pages are provided as to where to find the use of the MP; however, when viewing the pages, there is no additional guidance provided on the use of the MP.
• There is reference to MP1 in the student edition found on pages xxxii -- xlix, but the practice itself is not identified or referenced.
• MP2 is noted in the teacher edition to be located on pages xx – xxii in the student edition, but the pages are an index of “Connect Your Learning” activities.

The MP are found in the student edition on page viii, but there is no explanation of their meaning or importance.

The instructional materials reviewed for Grade 8 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials provide opportunities for students to construct viable arguments and analyze the arguments of others, but the instructions for those opportunities do not always prompt students to give explanations for their answers.

• There are opportunities for students to construct viable arguments and analyze the arguments of others in the content standards through check your understanding and homework exercises titled “Error Analysis” and “Writing in Math.”
• On page 51, students are asked to determine which work is correct and explain, but it misses the opportunity to explain the mistake.
• Under the section for MP3 on page T28, pages 17, 30, 51, 66, 120, 181, 198, 224, 225, 242 and 316 are listed and contain an “Error Analysis” problem. While students must analyze the problems to find mistakes, most of these problems require only procedural corrections.
• One exception is on page 267, in which students are asked to think about another student’s reasoning about rotation of a square and symmetry. Another exception is on page 312, in which students must provide a counterexample to prove the given student reasoning is not sound.
• Page 225 is also listed as having examples of the practice. Problem 23 is labeled Reasoning and asks the student if they can draw two triangles and prove them congruent with angle-angle-angle. They are to use their drawing to support their answer (of no, you can’t prove that) with no written explanation.
• In More Than One Way, students are supposed to “analyze and critique the solution plans of two students.” On pages 61, 124, 155, 196, 283 and 303, two different methods are explained step by step.  Students are asked to choose a method to use in a given problem. They are never asked why they chose a specific method or which would be more efficient in certain contexts or situations.
• Some sections have an “Error Analysis” exercise so that students can find and correct an error; however, they are told that there is an error to correct.

There are several times throughout the text (such as on page 155) that the materials present “more than one way” to solve a problem; however, they are given as examples. So, students only have to read through them instead of developing more than one way themselves and/or think through why both ways work.